For example, when we graph inequalities $(x^2+y^2-4)(y-x^2)<0$, we may pick one point that is not on the curves and see if the inequalities holds when we substitute the value of the point.
I wonder why this method works. How can one point represent the whole area including it?
The circle and parabola divide a plane into four regions. In each of them we have constant signs of both expressions. So, if our inequyality works for one point (lying not on the curves), then we have a desired configuration of the signs of both factors in the whole region represented by this point.